Research
Below are main topics of my research.
Sampling strategy / Discretization of measures with optimization
Sampled values of functions underlie many numerical methods like interpolation and quadrature.
To get accurate numerical approximation of functions and their functionals,
we need good points for sampling values of the functions.
I study methods for obtaining such points in various situation illustrated by the following papers.
In particular,
I'm interested in methods based on mathematical optimization of quantities evaluating point configurations.
They include logarithmic energies, their variants, maximum mean discrepancy, etc.
Representative papers

Ken'ichiro Tanaka, Alexis Akira Toda:
Discretizing Distributions with Exact Moments: Error Estimate and Convergence Analysis,
SIAM Journal on Numerical Analysis, Vol. 53, Issue 5 (2015), pp. 21582177 (doi:10.1137/140971269).
webpage

Ken'ichiro Tanaka, Masaaki Sugihara:
Design of accurate formulas for approximating functions in weighted Hardy spaces by discrete energy minimization,
IMA Journal of Numerical Analysis, 39(4) (2019), pp. 19571984 (doi: 10.1093/imanum/dry056).
webpage

Hiroaki Hirano, Ken'ichiro Tanaka:
Generation of collocation points in the method of fundamental solutions for 2D Laplace's equation,
JSIAM Letters Vol. 11 (2019), pp. 4952 (doi:10.14495/jsiaml.11.49).
webpage.

Ken'ichiro Tanaka:
Generation of point sets by convex optimization for interpolation in reproducing kernel Hilbert spaces,
Numerical Algorithms, 84 (2020), pp. 10491079 (doi:10.1007/s1107501900792w).
webpage

Ryunosuke Oshiro, Ken'ichiro Tanaka:
Effective methods for obtaining good points for quadrature in reproducing kernel Hilbert spaces,
JSIAM Letters Vol. 12 (2020), pp. 6164.
webpage

Toni Karvonen, Simo Sarkka, Ken'ichiro Tanaka:
Kernelbased interpolation at approximate Fekete points,
Numerical Algorithms Vol. 87 (2021), pp. 445468 (doi:10.1007/s1107502000973y).
webpage

Satoshi Hayakawa, Ken'ichiro Tanaka:
Monte Carlo construction of cubature on Wiener space,
Japan Journal of Industrial and Applied Mathematics, 39 (2022), pp. 543571. (doi: 10.1007/s13160021004966).
webpage

Kazuma Tsuji, Ken'ichiro Tanaka, Sebastian Pokutta:
Pairwise Conditional Gradients without Swap Steps and Sparser Kernel Herding,
Proceedings of the 39th International Conference on Machine Learning, PMLR 162 (2022), pp. 2186421883.
webpage of this paper /
webpage of the 39th ICML /
webpages of the videos of the presentations in the 39th ICML
Study of approximation formulas for holomorphic functions and their integrals
Holomorphic functions are fundamental in scientific computation.
I study numerical methods with variable transformations for holomorphic functions.
In particular,
I focus on methods with DoubleExponential (DE) transformations,
which were originally invented by Prof. Takahasi and Prof. Mori for quadrature.
Besides quadrature formulas with them,
I also study methods approximating functions with sinc funtions and their application.
They are collectively called sinc numerical methods.
Furthermore,
I have proposed improved approximation formulas by minimizing energies with external fields.
This study is also related to the above topic about sampling points.
Representative papers

Ken'ichiro Tanaka, Masaaki Sugihara, Kazuo Murota and Masatake Mori:
Function Classes for Double Exponential Integration Formulas.
Numerische Mathematik, Vol. 111 (2009), pp. 631655.
webpage

Ken'ichiro Tanaka, Masaaki Sugihara, and Kazuo Murota:
Function Classes for Successful DESinc Approximations.
Mathematics of Computation, Vol. 78 (2009), pp. 15531571.
webpage

Tomoaki Okayama, Ken'ichiro Tanaka, Takayasu Matsuo, Masaaki Sugihara:
DESinc methods have almost the same convergence property as
SESinc methods even for a family of functions fitting the SESinc methods
Part I: Definite integration and function approximation,
Numerische Mathematik, Vol. 125 (2013), pp. 511543 (doi: 10.1007/s002110130540x).
webpage

Ken'ichiro Tanaka, Tomoaki Okayama, Takayasu Matsuo, Masaaki Sugihara:
DESinc methods have almost the same convergence property as
SESinc methods even for a family of functions fitting the SESinc methods
Part II: Indefinite integration,
Numerische Mathematik, Vol. 125 (2013), pp. 545568 (doi: 10.1007/s0021101305419).
webpage

Ken'ichiro Tanaka, Masaaki Sugihara:
Design of accurate formulas for approximating functions in weighted Hardy spaces by discrete energy minimization,
IMA Journal of Numerical Analysis, 39(4) (2019), pp. 19571984 (doi: 10.1093/imanum/dry056).
webpage
Optimization of variable transformations for approximation formulas
This topic is concerned with the DE variable transformations in the above category.
The following papers present methods for improving the variable transformations.
In this study,
the SchwarzChristoffel transformations play an important role.
Representative papers

Shunki Kyoya, Ken'ichiro Tanaka:
Improvement of the double exponential formula with conformal maps based on the locations of singularities,
JSIAM Letters Vol. 11 (2019) pp. 6568 (doi: 10.14495/jsiaml.11.65).
webpage

Shunki Kyoya, Ken'ichiro Tanaka:
Construction of conformal maps based on the locations of singularities for improving the double exponential formula,
IMA Journal of Numerical Analysis, 40(4) (2020), pp. 27462776 (doi: 10.1093/imanum/drz040).
webpage
Numerical methods for economics and finance
The following papers are concerned with numerical methods related to economics and finance.
In particular,
the methods are related to
computation of functionals of stochastic processes used in economics and finance.
Besides the following papers,
there are dissertations of undergraduate and graduate students about this topic.
Representative papers

Ken'ichiro Tanaka, Alexis Akira Toda:
Discrete approximations of continuous distributions by maximum entropy,
Economics Letters, Vol. 118, Issue 3 (2013), pp. 445450.
webpage

Ken'ichiro Tanaka:
A fast and accurate numerical method for the symmetric Levy processes based on the Fourier transform and sincGauss sampling formula,
IMA Journal of Numerical Analysis 36(3) (2016), pp. 13621388 (doi:10.1093/imanum/drv038).
webpage

Satoshi Hayakawa, Ken'ichiro Tanaka:
Monte Carlo construction of cubature on Wiener space,
Japan Journal of Industrial and Applied Mathematics, 39 (2022), pp. 543571. (doi: 10.1007/s13160021004966).
webpage